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MEN  OF MATHEMATICS

points in the manifold, can be neglected. For simplicity we shall
state the definition when n = 4 - giving the distance between
two neighbouring points in a space of four dimensions: the
distance is the square root of

in which the ten coefficients gn, . . . , £34 are functions of
&i&s&ijCi* For a particular choice of the g's, one 'space* is
defined. Thus we might have glx = 1, £22 = *> £33 = 1? £« =
— 1, and all the other g's zero; or we might consider a space ia
which all the g's except g44 and g34 were zero, and so on. A space
considered in relativity is of this general kind hi which all the
£'s except ^11,^22^33^44 are zer°j an<i these are certain simple
expressions involving xltx^^x^
In the case of an n-dimensional space the distance between
neighbouring points is defined in a similar manner; the general
expression contains %n(n -{- 1) terms. The generalized Pytha-
gorean formula for the distance between neighbouring points
being given, it is a solvable problem ua the integral calculus to
find the distance between any two points of the space. A space
whose metric (system of measurement) is defined by a formula
of the type described is called Riemannian.
Curvature, as conceived by Riemann (and before him by
Gauss; see chapter on the latter) is another generalization from
common experience. A straight line has zero curvature; the
'measure' of the amount by which a curved line departs from
straightness may be the same for every point of the curve (as it
is for a circle), or it may vary from point to point of the curve,
when it becomes necessary again to express the 'amount of
curvature' through the use of infinitesimals. For curved sur-
faces, the curvature is measured similarly by the amount of
departure from a plane, which has zero curvature. This may be
generalized and made a little more precise as follows. For sim-
plicity we state first the situation for a two-dimensional space,
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